MATLAB: Can someone provide me the theory and math behind this function of eigen

eigenvalueseigenvector

[V,D] = eig(A,B)

returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors, so that A*V = B*V*D.

Best Answer

for each column number j,

A*V = B*V*D

implies

A*xj = lambdaj*B*xj

where

xj = V(:,j)
lambdaj = D(j,j)

This is just a generalization of normal eigen value problem.

A*xj = lambdaj*xj

If B is invertible, V and D is the same as standard eigen vectors/values of M := inv(B)*A.

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help eig

    E = eig(A,B) is a vector containing the generalized eigenvalues
    of square matrices A and B.
    [V,D] = eig(A,B) produces a diagonal matrix D of generalized
    eigenvalues and a full matrix V whose columns are the
    corresponding eigenvectors so that A*V = B*V*D.

Hope this helps.

Thank you for formally accepting m answer.

Greg

MATLAB: Question about number format

However, this is not the correct answer

You're assuming there is only one correct answer. That is not a valid assumption in this case.

Multiplying the eigenvector by any non-zero scalar just scales the eigenvector, and that scaled eigenvector still satisfies the equation that eigenvectors must satisfy. This makes sense if you look at the essential definition according to Wikipedia.

"In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue."

See this example:

% Compute eigenvalues and eigenvectors
% A = magic(6)
[V, D] = eig(A)
% Does the first eigenvalue and eigenvector satisfy A*V = V*D?
shouldBeCloseToZeroVector1 = A*V(:, 1) - V(:, 1) * D(1, 1) % Close enough
% Multiple the first eigenvector by 2
twice = 2*V(:, 1);
% Does two times the first eigenvalue and eigenvector satisfy A*V = V*D?
shouldBeCloseToZeroVector2 = A*twice - twice * D(1, 1) % Also close enough

This is not a bug.

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